0:00:13 | thank you um good afternoon |
---|---|

0:00:15 | um |

0:00:15 | so |

0:00:16 | one of the problem that we tried to fall an image processing is a a is of some data |

0:00:21 | that have been exposed to geometric transformations |

0:00:24 | for example we might want to reduce is such data are or or classify them in a transformation invariant like |

0:00:31 | one of a common approach |

0:00:32 | for um be with such problems is the use all my fault models |

0:00:36 | so in this work we have a um concentrated on that transformation manifold |

0:00:42 | and it at transformation manifold in that |

0:00:44 | family a little images that are generated by a in certain set of a geometric transformations to reference better |

0:00:50 | for example if you take the structure ten P |

0:00:53 | we do not its transformation manifold by |

0:00:56 | P here |

0:00:57 | and uh so we assume that this is an an picks out much and the mind fold is also a |

0:01:02 | subset of R and |

0:01:03 | this case |

0:01:04 | each each which on this the transformation manifold is a geometric a transformed version of P |

0:01:10 | and we define just transformation by a parameter vector or that in the parameter space |

0:01:16 | and just that uh this uh a lot that house as the type of the geometric transformation for instance uh |

0:01:21 | it could be any combination of two D to transformations like |

0:01:25 | rotation and translation scale change one can find also for example |

0:01:32 | so our for |

0:01:33 | as in this work is uh the following we assume that we in one |

0:01:37 | a set of uh geometric metric "'cause" transform observations of a signal type like uh a five digit |

0:01:44 | illustration |

0:01:45 | a from the observations we are trying to construct a |

0:01:49 | it pattern transformation manifold so in part we want to find a pattern P |

0:01:53 | such that the transformation manifold of P |

0:01:56 | uh represents about this state that so it's like a extra fitting problem but if it and when you to |

0:02:01 | the data instead of like |

0:02:04 | so that |

0:02:05 | problem is the to find a spectral P |

0:02:09 | so um |

0:02:10 | this kind of a framework has some meaning that applications for the modeling and the registration of the input data |

0:02:16 | on |

0:02:16 | including is also possible because we will be finding the pattern P in terms of uh some parametric at so |

0:02:22 | it's also used to called the input data and like |

0:02:26 | and also another ad don't use that we provide an unknown not to the model for our money false |

0:02:30 | so that we can since to sides we can generate a new data on the manifold |

0:02:34 | and this makes it possible to compute |

0:02:37 | exactly at distance |

0:02:38 | bit mean it's estimates and construct a old |

0:02:41 | so this can be a a time as some classification settings for instance if we are given that test image |

0:02:45 | some geometric transformation if you want to class by |

0:02:48 | we just need to compute its distance to the uh of the transformation manifold |

0:02:55 | so for a so that all i was first uh try to form like the problem than i will describe |

0:03:00 | a solution that of the based on computing a representative pattern P uh with the greedy out to great really |

0:03:06 | by selecting some atoms from a parametric dictionary |

0:03:10 | so uh here's a show the manifold is not the by a and he and each image on the manifold |

0:03:15 | of this from by you um that P it means a |

0:03:18 | and the pattern P and uh i applied it to from some which the on it |

0:03:22 | and we denote are |

0:03:24 | uh input them just by you why this R |

0:03:26 | uh uh this gone for some geometric transformations |

0:03:29 | and what we are trying to do is find a common reference pattern P |

0:03:33 | and model be um input points |

0:03:36 | uh uh is transformation of this common pattern P |

0:03:39 | plus some uh ever try and this error time you i shows the deviation of |

0:03:44 | the image you i from the construct mind for |

0:03:47 | and uh we assume that you know the type of transformations for instance you know bidders rotation translation scale that's |

0:03:53 | a drought but |

0:03:54 | still we to re just their input that the that means we need to compute a vector along the i |

0:03:58 | for each of input image |

0:04:01 | and then we use this idea phone construct thing P is a combination of some uh |

0:04:06 | i |

0:04:07 | so P equals the sum of atoms a J base of by this collection of C J |

0:04:12 | and we also assume that use that sums come from a parametric dictionary that means |

0:04:16 | each atom in a dictionary |

0:04:18 | is a a geometrically transformed version of an an i'm not function so mother functions from by five here |

0:04:24 | this is a a a a a marshal so the geometric transformation |

0:04:28 | and some possible a little uh some examples for this on and will uh a generating mother function could be |

0:04:33 | a process cost and motor function or |

0:04:36 | an isotropic refinement but or functions from by a and R |

0:04:39 | and here you see some at some that are um the i form house thousand motor function to some geometric |

0:04:45 | transformation |

0:04:47 | and um here is the formation of this month for fitting problem |

0:04:51 | so we like to minimize the total distance of our input images to construct the money full we shall we |

0:04:56 | by |

0:04:57 | E |

0:04:57 | and and we want to we would like to uh it she'll just by picking a subset |

0:05:02 | all the atoms in the dictionary slow not P us these A J that comes a G R and |

0:05:07 | also optimized the for options of these atoms |

0:05:10 | such that this total distance that are he is mean |

0:05:15 | the uh and you know next case of this uh read out from that we propose |

0:05:19 | so we first so choose arbitrarily and that to mean the dictionary |

0:05:24 | a suitable one and then be set that part pattern P |

0:05:28 | uh and then we compute the projection of are input images on the money |

0:05:33 | and then here the main loop now all uh at each iteration we select and at some at a and |

0:05:38 | the coefficients C |

0:05:39 | such that we reduce the errors |

0:05:42 | and then we at this at some our pattern |

0:05:46 | so this this based on my fault |

0:05:48 | and an now the money for that it is a very compute the projections of are uh input image of |

0:05:53 | on them i if what and then we continue this loop |

0:05:56 | and till the the data approximation error is minimal |

0:06:00 | and now how to be a of the minimisation of this error are still i'm fortunes as error has a |

0:06:04 | complicated the panels on the at and option |

0:06:07 | and is for the following reason uh let's imagine that we are now in the j-th iterations of the already |

0:06:12 | have a computer this manifold and P J A lines one |

0:06:15 | and so if you take an input image you why i mean its projection |

0:06:20 | smile of that's already compute so we know the parameter vector or but i corresponding |

0:06:24 | i mean were when the a minor followed by adding and you want |

0:06:28 | it's projection point no change |

0:06:30 | and most probably will correspond to a parameter vector number i pride which is a a different from um by |

0:06:36 | and we don't know what this number by prime |

0:06:38 | will be |

0:06:39 | uh but if we right down the total distance used in that it depends on this uh a will real |

0:06:45 | new you of the parameter vector by prior so |

0:06:47 | that's this it's not use it to um |

0:06:49 | minimize directly this uh error E |

0:06:52 | so we uh defined an approximation you have |

0:06:55 | of of know instead of we minimize this you |

0:06:58 | and then what is the C had it is just the sum of the kind and this distance a little |

0:07:02 | imp point to the new my fall |

0:07:04 | and and time and this as as as follows we had a new manifold now and we obtain a first |

0:07:09 | order approximation of this money there on the projection points that are already or |

0:07:14 | and then the change in the sense of you i for this manifold is just the this distance between you |

0:07:19 | Y and a |

0:07:20 | uh a first order approximation |

0:07:23 | so uh |

0:07:24 | actually be do something pretty straightforward to minimize the that we just to each of the atoms of addiction or |

0:07:29 | one by one |

0:07:30 | and for each at an we find we compute the optimum options see that minimize the stereo tab |

0:07:36 | and if we you right this you had as a function of C |

0:07:39 | um we see that is a it's in the form of a racial function that means this function at a |

0:07:44 | i and G I's are on my meals of C |

0:07:47 | so in general um |

0:07:49 | such a function |

0:07:50 | has several local minima |

0:07:52 | and it where we can seen in practice a experiments we have seen that it is also in most most |

0:07:57 | of the time is possible to minimise you that just by a simple a and the sound out or two |

0:08:01 | is not that |

0:08:02 | extreme complicated function in practice |

0:08:05 | so um we try |

0:08:06 | each at and can compute all the local options uh and then in of all the atoms if we the |

0:08:11 | best one |

0:08:12 | that you small star |

0:08:14 | then we add the this at some to the new cut and uh by its uh optimal corruption |

0:08:19 | and you repeat the use of course |

0:08:22 | so now um some experiments for some on for and a later on |

0:08:27 | and in this experiment we use a transformation model of of uh we use the transformation manifold model of the |

0:08:32 | mansion three so we have uh |

0:08:35 | rotation and then it would be to two national translation |

0:08:38 | uh so we can generate a the syntactical path and by adding some loss in and a and are i |

0:08:43 | don't |

0:08:44 | and uh so we construct a different data sets from this at some uh each dataset consists of some random |

0:08:50 | geometric transformations of this the synthetic that pattern |

0:08:53 | and you have a four out to each data of that that uh it it is a uh gaussian noise |

0:08:57 | with |

0:08:58 | for noise variances for sports data set |

0:09:00 | and we use that dictionary consisting of some cost in them the R |

0:09:05 | so um here you see the data approximation error or lot that just like the noise variance |

0:09:10 | so i approximation error is the total squared distance of input images |

0:09:14 | the computed my |

0:09:15 | is see that it's uh it is it has a a linear variation |

0:09:19 | like to noise variance which is an expected result |

0:09:22 | uh uh have are if you pay attention here does just line doesn't pass from the origin so this actually |

0:09:28 | re we'll the error of the algorithm |

0:09:30 | and there are two main source of though |

0:09:32 | uh for this error of all is that use a grid out them and it doesn't have an optimal performance |

0:09:37 | T |

0:09:38 | and secondly we use a dictionary of |

0:09:40 | fine size |

0:09:41 | that's the discrete or this also introduce some there |

0:09:46 | and uh experiment sometime in it |

0:09:49 | this time we use the four dimensional transformation model because we also have a you changed um in the |

0:09:55 | um a as |

0:09:56 | and is they are uh we use a hundred to the geometric to transforms |

0:10:01 | hundred five |

0:10:03 | and use a similar dictionary so on the left you see some of the sound of they in the experiment |

0:10:08 | and on the right so uh you see the patch that we obtain the twenty four at |

0:10:13 | so it looks like a five digit that sure about the characters |

0:10:16 | digits five um despite the variation |

0:10:20 | the they does that |

0:10:22 | and also uh some uh for some numerical comparison we have compared to some rec |

0:10:26 | approach |

0:10:27 | and we have use this error measure a measure which is a the data approximation error |

0:10:32 | so in the first to uh a reference is that have again computed |

0:10:37 | progressive approximations of the uh are designed |

0:10:40 | so in the first one we have applied matching force on a typical are in the data that the average |

0:10:46 | are here and we have chosen it to be |

0:10:48 | the input data out it close as |

0:10:50 | to the centroid of all and i say |

0:10:52 | J |

0:10:55 | and uh in the second one we have applied simultaneous matching pursuit on or a line |

0:11:00 | to achieve that |

0:11:01 | sparse uh |

0:11:02 | find |

0:11:03 | i |

0:11:04 | and we don't |

0:11:06 | and finally as order approach like everyone provide a comparison between our method and uh |

0:11:12 | classical manifold learning |

0:11:14 | and it doesn't on that in some of the typical manifold learning algorithms they make use of the assumption that |

0:11:19 | data has a local in your be or on the mind |

0:11:22 | so we just uh a compute the this uh a local linear manifold approximation error |

0:11:27 | is the sum of |

0:11:28 | these |

0:11:29 | E i one E i |

0:11:30 | is |

0:11:31 | uh the distance between a point you Y |

0:11:34 | and the plane thing from the nearest neighbor |

0:11:39 | oh um you see that are lots here are so the move of is the transformation invariant matching proof of |

0:11:46 | word that we have proposed so we get the best or performance |

0:11:50 | um we see that the red of corresponds to matching pursuit on average but |

0:11:54 | a if i and it's as that |

0:11:56 | okay so to do that and the data that that that for all |

0:12:00 | uh |

0:12:00 | you know like you're |

0:12:02 | i that that that would be a lot but |

0:12:04 | is it or not |

0:12:06 | and this station is and that the one time and the patterns are um |

0:12:10 | when we have applied simultaneous |

0:12:12 | a a sparse |

0:12:13 | estimation of that |

0:12:14 | such P |

0:12:17 | and finally some experiments on |

0:12:19 | face image this time |

0:12:20 | this time at high dimensional the because we have an an isotropic scaling |

0:12:24 | and we have used some |

0:12:26 | um face images of the same subject but we also uh i had some |

0:12:30 | but uh in the data set and some variation of facial expression that we don't not model |

0:12:36 | things like uh a facial expression variations but |

0:12:38 | these things are are rather close there that the source of the deviation from the computed manifold |

0:12:44 | and uh uh here on the right so the that some they like can from the data set of on |

0:12:48 | the right to face them me that we have computed |

0:12:50 | so it looks |

0:12:51 | more or less like the phase of the same person |

0:12:53 | there is also some kind of averaging and |

0:12:55 | facial expression and uh |

0:12:57 | we you have a doubt that all lesions |

0:13:00 | and um if you look at the error loss we see here that |

0:13:04 | so okay K even if is still get the best error for from a uh we can see here that |

0:13:08 | the and and in is some people's the perform about |

0:13:11 | this is because the number of variation |

0:13:13 | then the face image of the same person are |

0:13:16 | what's smaller and compared to the micro variation the hand |

0:13:19 | it |

0:13:21 | that typical people pattern of the data set |

0:13:23 | like to approximate that all patterns |

0:13:26 | i mean there and if you look at this uh that line as locally in or approximation or is pretty |

0:13:31 | i |

0:13:32 | and very for this is that the data uh do we have just use thirty five of just so the |

0:13:36 | data is sparse the sample on the my fault |

0:13:39 | the local linearity assumption that hold the anymore |

0:13:42 | you had to |

0:13:45 | so um to a little bit have present presented the method to the for transformation and rent sparse approximation of |

0:13:51 | a set of signals |

0:13:52 | we are we have built a representative pattern with the grid out some by a parametric atom selection |

0:13:58 | and the complexity of the matter a method that we propose a |

0:14:01 | changes linearly with respect to the number of atoms in the dictionary |

0:14:06 | as a linear with respect to the number of images and the input that |

0:14:09 | and it has a corn of the panels on the notion of the mind for the image resolution |

0:14:15 | a there are um we have shown in another work that |

0:14:18 | uh under some assumptions on the transformation model |

0:14:22 | and also the structure of the dictionary we can it cheap a joint optimization of the at parameters |

0:14:28 | and uh the functions C |

0:14:29 | so in this case uh we optimize on the continuous dictionary might of fall rather than a |

0:14:36 | um |

0:14:36 | fixed dictionary a |

0:14:37 | speech uh at samples |

0:14:40 | and in this case uh we get rid of just |

0:14:42 | for star here we don't have a uh a the depends a number of because the local jurisdiction |

0:14:50 | so um is a |

0:14:51 | final remark a um are right |

0:14:54 | can related to to as in general one is sparse signal approximation of and the other is a all learning |

0:15:00 | so um what's that we gained over sparse signal approximation at like and P S and E |

0:15:05 | it is that we H you uh in a variance to geometric transformations of the data of you you we |

0:15:10 | use a transformation manifold model |

0:15:13 | on the other hand the and on to as we have over classical month learning algorithms are the following |

0:15:18 | a first of all we provide an article model for the data and that has a nice properties like a |

0:15:23 | it's the french it's move |

0:15:25 | it is also used to call the take the |

0:15:28 | parametric atoms |

0:15:29 | it a L the end generation need they on the manifold |

0:15:32 | and finally it has that it can still work if uh the something of that database |

0:15:37 | sparse |

0:15:38 | whereas as um |

0:15:39 | many need fall that work and would require a much |

0:15:42 | source |

0:15:43 | oh |

0:15:44 | so uh that's all and take you very much for function |

0:15:52 | thank you |

0:15:54 | as a first |

0:16:05 | i |

0:16:05 | yeah that's it the best to extract that um |

0:16:09 | for time read |

0:16:10 | actually what we do is we on |

0:16:13 | minimize mean Z V in one as an approximation yeah |

0:16:17 | so we do a oh so at each iteration okay we minimize the if that's that's T |

0:16:22 | but as E that is not equal to you that's one reason the second reason |

0:16:26 | um |

0:16:28 | so it and if you mean my is |

0:16:31 | this is one of the projection points change |

0:16:34 | the forty two reasons uh a menu do this optimization thing you want to a guarantee that you will reduced |

0:16:40 | so but we do in practice of that |

0:16:42 | uh okay so we try this pick the best that some of we want to the project and than on |

0:16:46 | be check if there are set of it just um we are fine accounting a if the error you don't |

0:16:51 | the green |

0:16:52 | the we try and reckon at them like don't to pick the best one but pick the second best one |

0:16:57 | and then tried |

0:16:58 | a but we we all |

0:16:59 | well of course they are able to uh |

0:17:03 | but a set up a date |

0:17:04 | only if the error is it just so |

0:17:06 | since the V we reduced the error E he for sure and in each iteration and so uh it has |

0:17:12 | a lower bound and that it has to converge at some point |

0:17:15 | um |

0:17:16 | oh for situation |

0:17:21 | what do you |

0:17:24 | we |

0:17:26 | exploit |

0:17:28 | yeah i i that um |

0:17:29 | i i think in whatever may you define fine of i mean whatever kind of transformation you can there i |

0:17:36 | think as long as a um you did find this error E and this like like double distance of them |

0:17:40 | but that you for it |

0:17:42 | to the degree that each iteration |

0:17:44 | um |

0:17:45 | yeah so if you use degrees and a function that is lower bound that uh it means that a test |

0:17:50 | to code word after a while |

0:17:52 | is monotonically decreasing function |

0:17:54 | you |

0:17:57 | as a in |

0:18:01 | i seven it depends on also a should be you have to to be used for the the dictionary |

0:18:12 | you you need a note the is actually that it to to play even if you do the meeting |

0:18:18 | you try to would |

0:18:21 | yeah like your that that's it |

0:18:24 | so um so that it's a question about dictionary learning i guess um we have a on anything like a |

0:18:30 | um |

0:18:31 | i mean doing something like a C you like case we get to optimize that |

0:18:35 | one reason for this is that we really would like to |

0:18:38 | to to been a parametric forms of all |

0:18:41 | uh a we need them to be differentiable function because we're talking about ten just to the might so they |

0:18:47 | just can't be an arbitrary function |

0:18:49 | so that than this a but uh this think that i have mentioned here |

0:18:53 | uh |

0:18:54 | finally this |

0:18:56 | the for all |

0:18:57 | that |

0:18:58 | kind of such as this field of addiction learning because here we have a a dictionary of money for and |

0:19:03 | not what we do is |

0:19:04 | you optimized |

0:19:06 | a on the big show mind fall that are you optimise the parameters of the atoms |

0:19:10 | this is |

0:19:12 | can related to a lot |

0:19:14 | but we i consider a differentiable uh |

0:19:18 | at like a in a and i to be any the french on article function so it's gender can that's |

0:19:24 | yeah but is not learn from the they are no we with that |

0:19:28 | yeah |

0:19:29 | as wish to |

0:19:33 | and that was to uh you said that yeah actually used to as the fact that the sparse approach |

0:19:39 | but the a D do actually they will go explicitly use is the constraint in your uh optimization |

0:19:47 | so yeah and question is there a house as is it depending on your T and know how were how |

0:19:52 | do you think this into account the in to you are uh |

0:19:55 | uh optimization problem |

0:19:57 | we have introduced the |

0:19:59 | that and L one norm or or or or no we don't take it this like to that |

0:20:04 | hmmm |

0:20:04 | you oh |

0:20:06 | sparse sparse the they'd sure talking about is and which main |

0:20:09 | so here and uh we there's sparsity in a a times of these |

0:20:13 | dictionary atoms that we use so we have to |

0:20:16 | um |

0:20:17 | version |

0:20:25 | yeah so here |

0:20:26 | uh uh you have |

0:20:28 | J O D is that some so if K is much smaller and and number of cells that you have |

0:20:33 | and the in which is done this pattern P is |

0:20:35 | sparse in this domain just consisting of or |

0:20:38 | a or a an hour |

0:20:39 | and |

0:20:39 | so um the made that you stick here is that look okay you can do that like |

0:20:44 | okay and not take uh fifty atoms |

0:20:46 | i keep the best fifty the atoms and um |

0:20:49 | use yeah head and it's |

0:20:50 | parsons and |

0:20:51 | okay approximation |

0:20:53 | yes |

0:20:55 | okay |

0:20:57 | as question |

0:20:59 | a you've not that again |

0:21:02 | a |

0:21:03 | no |

0:21:04 | and |